Go read the general rules on invoking powers, RFTG rules page 4, and the GS rules, page 11, which discusses the timing of this specific power.
If you have zero cards in hand with Rebel Cantina + Damaged Alien Factory on the tableau, are you allowed to produce a good on the DAF or not? (In other words, do you draw the card from the RC first, to be able to discard it to the DAF's produce second?)
If you have no cards in hand and Gem World + Damaged Alien Factory on the tableau, are you allowed to produce on the DAF? (Do you draw the card from the GW first?)
Yes, if you choose to invoke the RC power or Gem World power first.
If you have no cards in hand and Rebel Sympathizers + Damaged Alien Factory on the tableau [and either called Produce or have a Produce on Windfall power], are you allowed to [use your Produce bonus or Produce on Windfall power first to produce on RS and then use the card gained to] produce on the DAF?
If you have no cards in hand and Research Labs + Damaged Alien Factory on the tableau, are you allowed to produce on the DAF? (Do you draw the card from the RL first?)
Of course not. You cannot use future effects to fuel another power in a circular fashion. Powers are invoked *sequentially*. If you invoke the RL power first, then you don't draw anything because you haven't produced any Alien goods. If you invoke DAF first, you have no card to discard to produce an Alien good.
In addition to these cases, the Diversified Economy example given in the GS rules illustrates the "a power cannot interrupt another power" timing rule.